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May 17th, 2017, 02:35 AM   #1
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Differentiable function

Let $f: R -> R $ be a twice differentiable function. Then which of the following is true ?
A) If $f(0)=0= f''(0) $ then $f'(0)=0$.
B) $f$ is a polynomial.
C) $f'$ is continuous.
D) If $f''(x) > 0$ for all $x$ in $R$ then $f(x)>0$ for all $x$ in $R$.
I have checked some examples for the option A and it is satisfying this condition. So I guess the answer is A.
Please let me know if I'm wrong
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May 17th, 2017, 04:25 AM   #2
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Does choice A hold true for $f(x)=\sin{x}$ ?
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May 17th, 2017, 04:52 AM   #3
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Quote:
Originally Posted by skeeter View Post
Does choice A hold true for $f(x)=\sin{x}$ ?
No...
Thanks!
I only checked with polynomials I guess
So any guess what could be the correct option ?
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May 17th, 2017, 04:56 AM   #4
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Quote:
Originally Posted by skeeter View Post
Does choice A hold true for $f(x)=\sin{x}$ ?
So i think it should be continuous to be twice differentiable... right ?
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May 17th, 2017, 05:57 AM   #5
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Yes. A necessary condition for the derivative of a function to exist (at a point) is that the function be continuous (at that point). If we $f$ is twice differentiable, then $f'$ is differentiable and thus continuous.

Almost all functions we look at (especially in learning differentiation) are infinitely (also known as "continuously") differentiable. But not all functions are. The classic example is $$f(x)=\begin{cases} x^2\sin\frac1x &(x \ne 0) \\ 0 & (x=0)\end{cases}$$ which is differentiable only once with $f'$ being disc9ntinuous at $x=0$.
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